 |
Similarly for the fifteen similar Propositions, in terms of x and m, or of y and m.
These thirty-two Propositions of Relation are the only ones that we shall have to represent on this Diagram.
The Reader should now get his genial friend to question him on the following four Tables.
The Victim should have nothing before him but a blank Triliteral Diagram, a Red Counter, and 2 Grey ones, with which he is to represent the various Propositions named by the Inquisitor, e.g. "No y' are m", "Some xm' exist", &c., &c. pg046 TABLE V.
. -. . -. Some xm exist . -. = Some x are m . - -. = Some m are x . - -. (.) ( ) ( ) - - - - . -. - - - - No xm exist . - -. = No x are m . - -. = No m are x . -. . -. - . -. Some xm' exist . -. (.) = Some x are m' ( ) ( ) . - -. = Some m' are x . - -. - - - - . -. - - - - No xm' exist . - -. = No x are m' . - -. = No m' are x . -. . -. - . -. Some x'm exist . -. = Some x' are m . - -. = Some m are x' . - -. - - - - . -. - - - - (.) No x'm exist ( ) ( ) . - -. = No x' are m . - -. = No m are x' . -. . -. - . -. Some x'm' exist . -. = Some x' are m' . - -. = Some m' are x' . - -. - - - - . -. - - - - No x'm' exist . - -. = No x' are m' . - -. (.) = No m' are x' ( ) ( ) . -. . -. . -. pg047 TABLE VI.
. -. . -. Some ym exist . -. = Some y are m . - -. = Some m are y . - -. ( ) - (.) - - . -. - - - - No ym exist ( ) . - -. = No y are m . - -. = No m are y . -. . -. - . -. Some ym' exist . -. = Some y are m' ( ) . - -. = Some m' are y . - -. (.) - - - . -. - - - - No ym' exist . - -. = No y are m' . - -. = No m' are y ( ) . -. . -. - . -. Some y'm exist . -. = Some y' are m . - -. = Some m are y' . - -. ( ) - - (.) - . -. - - - - No y'm exist ( ) . - -. = No y' are m . - -. = No m are y' . -. . -. - . -. Some y'm' exist . -. = Some y' are m' ( ) . - -. = Some m' are y' . - -. - - - (.) . -. - - - - No y'm' exist . - -. = No y' are m' . - -. = No m' are y' ( ) . -. . -. . -. pg048 TABLE VII.
. -. . -. . -. ( ) ( ) All x are m (.) . - -. . - -. (.) ( ) ( ) - - - - . -. - - - - . - -. All x are m' . - -. . -. . -. - . -. . -. All x' are m . - -. . - -. - - - - . -. - - - - (.) ( ) ( ) . - -. All x' are m' . - -. ( ) ( ) (.) . -. . -. - . -. . -. All m are x . - -. . - -. (.) ( ) ( ) - - - - . -. - - - - ( ) ( ) (.) . - -. All m are x' . - -. . -. . -. - . -. . -. (.) All m' are x ( ) ( ) . - -. . - -. - - - - . -. - - - - . - -. All m' are x' . - -. ( ) ( ) (.) . -. . -. . -. pg049 TABLE VIII.
. -. . -. . -. ( ) All y are m . - -. . - -. ( ) - (.) - - . -. (.) - - - ( ) . - -. All y are m' . - -. ( ) . -. . -. - . -. . -. ( ) All y' are m . - -. . - -. ( ) - - (.) - . -. - - - (.) ( ) . - -. All y' are m' . - -. ( ) . -. . -. - . -. . -. All m are y . - -. . - -. ( ) ( ) - (.) - - . -. - - (.) - ( ) ( ) . - -. All m are y' . - -. . -. . -. - . -. . -. ( ) All m' are y ( ) . - -. . - -. (.) - - - . -. - - - (.) . - -. All m' are y' . - -. ( ) ( ) . -. . -. . -.
pg050
CHAPTER III.
REPRESENTATION OF TWO PROPOSITIONS OF RELATION, ONE IN TERMS OF x AND m, AND THE OTHER IN TERMS OF y AND m, ON THE SAME DIAGRAM.
The Reader had better now begin to draw little Diagrams for himself, and to mark them with the Digits "I" and "O", instead of using the Board and Counters: he may put a "I" to represent a Red Counter (this may be interpreted to mean "There is at least one Thing here"), and a "O" to represent a Grey Counter (this may be interpreted to mean "There is nothing here").
The Pair of Propositions, that we shall have to represent, will always be, one in terms of x and m, and the other in terms of y and m.
When we have to represent a Proposition beginning with "All", we break it up into the two Propositions to which it is equivalent.
When we have to represent, on the same Diagram, Propositions, of which some begin with "Some" and others with "No", we represent the negative ones first. This will sometimes save us from having to put a "I" "on a fence" and afterwards having to shift it into a Cell.
[Let us work a few examples.
(1)
"No x are m'; No y' are m".
Let us first represent "No x are m'". This gives us Diagram a.
Then, representing "No y' are m" on the same Diagram, we get Diagram b. pg051 a b . -. . -. (O) (O) (O) (O) . - -. . - -. (O) - - - - - - - - (O) . - -. . - -. . -. . -.
(2)
"Some m are x; No m are y".
If, neglecting the Rule, we were begin with "Some m are x", we should get Diagram a.
And if we were then to take "No m are y", which tells us that the Inner N.W. Cell is empty, we should be obliged to take the "I" off the fence (as it no longer has the choice of two Cells), and to put it into the Inner N.E. Cell, as in Diagram c.
This trouble may be saved by beginning with "No m are y", as in Diagram b.
And now, when we take "Some m are x", there is no fence to sit on! The "I" has to go, at once, into the N.E. Cell, as in Diagram c.
a b c . -. . -. . -. . - -. . - -. . - -. (I) (O) (O) (I) - - - - - - - - - - - - (O) (O) . - -. . - -. . - -. . -. . -. . -.
(3)
"No x' are m'; All m are y".
Here we begin by breaking up the Second into the two Propositions to which it is equivalent. Thus we have three Propositions to represent, viz.—
(1) "No x' are m'; (2) Some m are y; (3) No m are y'".
These we will take in the order 1, 3, 2.
First we take No. (1), viz. "No x' are m'". This gives us Diagram a. pg052 Adding to this, No. (3), viz. "No m are y'", we get Diagram b.
This time the "I", representing No. (2), viz. "Some m are y," has to sit on the fence, as there is no "O" to order it off! This gives us Diagram c.
a b c . -. . -. . -. . - -. . - -. . - -. (O) (O) - - - - - - - - - (I) - - (O) (O) . - -. . - -. . - -. (O) (O) (O) (O) (O) (O) . -. . -. . -.
(4)
"All m are x; All y are m".
Here we break up both Propositions, and thus get four to represent, viz.—
(1) "Some m are x; (2) No m are x'; (3) Some y are m; (4) No y are m'".
These we will take in the order 2, 4, 1, 3.
First we take No. (2), viz. "No m are x'". This gives us Diagram a.
To this we add No. (4), viz. "No y are m'", and thus get Diagram b.
If we were to add to this No. (1), viz. "Some m are x", we should have to put the "I" on a fence: so let us try No. (3) instead, viz. "Some y are m". This gives us Diagram c.
And now there is no need to trouble about No. (1), as it would not add anything to our information to put a "I" on the fence. The Diagram already tells us that "Some m are x".]
a b c . -. . -. . -. (O) (O) . - -. . - -. . - -. (I) - - - - - - - - - - - - (O) (O) (O) (O) (O) (O) . - -. . - -. . - -. (O) (O) . -. . -. . -.
[Work Examples Sec. 1, 9-12 (p. 97); Sec. 2, 1-20 (p. 98).]
pg053
CHAPTER IV.
INTERPRETATION, IN TERMS OF x AND y, OF TRILITERAL DIAGRAM, WHEN MARKED WITH COUNTERS OR DIGITS.
The problem before us is, given a marked Triliteral Diagram, to ascertain what Propositions of Relation, in terms of x and y, are represented on it.
The best plan, for a beginner, is to draw a Biliteral Diagram alongside of it, and to transfer, from the one to the other, all the information he can. He can then read off, from the Biliteral Diagram, the required Propositions. After a little practice, he will be able to dispense with the Biliteral Diagram, and to read off the result from the Triliteral Diagram itself.
To transfer the information, observe the following Rules:—
(1) Examine the N.W. Quarter of the Triliteral Diagram.
(2) If it contains a "I", in either Cell, it is certainly occupied, and you may mark the N.W. Quarter of the Biliteral Diagram with a "I".
(3) If it contains two "O"s, one in each Cell, it is certainly empty, and you may mark the N.W. Quarter of the Biliteral Diagram with a "O". pg054 (4) Deal in the same way with the N.E., the S.W., and the S.E. Quarter.
[Let us take, as examples, the results of the four Examples worked in the previous Chapters.
(1) . -. (O) (O) . - -. (O) - - - - (O) . - -. . -.
In the N.W. Quarter, only one of the two Cells is marked as empty: so we do not know whether the N.W. Quarter of the Biliteral Diagram is occupied or empty: so we cannot mark it.
. -. (O) - - . -.
In the N.E. Quarter, we find two "O"s: so this Quarter is certainly empty; and we mark it so on the Biliteral Diagram.
In the S.W. Quarter, we have no information at all.
In the S.E. Quarter, we have not enough to use.
We may read off the result as "No x are y'", or "No y' are x," whichever we prefer.
(2) . -. . - -. (O) (I) - - - - (O) . - -. . -.
In the N.W. Quarter, we have not enough information to use.
In the N.E. Quarter, we find a "I". This shows us that it is occupied: so we may mark the N.E. Quarter on the Biliteral Diagram with a "I".
. -. (I) - - . -.
In the S.W. Quarter, we have not enough information to use.
In the S.E. Quarter, we have none at all.
We may read off the result as "Some x are y'", or "Some y' are x", whichever we prefer. pg055 (3) . -. . - -. (O) - (I) - - (O) . - -. (O) (O) . -.
In the N.W. Quarter, we have no information. (The "I", sitting on the fence, is of no use to us until we know on which side he means to jump down!)
In the N.E. Quarter, we have not enough information to use.
Neither have we in the S.W. Quarter.
. -. - - (O) . -.
The S.E. Quarter is the only one that yields enough information to use. It is certainly empty: so we mark it as such on the Biliteral Diagram.
We may read off the results as "No x' are y'", or "No y' are x'", whichever we prefer.
(4) . -. (O) . - -. (I) - - - - (O) (O) . - -. (O) . -.
The N.W. Quarter is occupied, in spite of the "O" in the Outer Cell. So we mark it with a "I" on the Biliteral Diagram.
The N.E. Quarter yields no information.
. -. (I) - - . -.
The S.W. Quarter is certainly empty. So we mark it as such on the Biliteral Diagram.
. -. (I) - - (O) . -.
The S.E. Quarter does not yield enough information to use.
We read off the result as "All y are x."]
[Review Tables V, VI (pp. 46, 47). Work Examples Sec. 1, 13-16 (p. 97); Sec. 2, 21-32 (p. 98); Sec. 3, 1-20 (p. 99).]
pg056
BOOK V.
SYLLOGISMS.
CHAPTER I.
INTRODUCTORY
When a Trio of Biliteral Propositions of Relation is such that
(1) all their six Terms are Species of the same Genus,
(2) every two of them contain between them a Pair of codivisional Classes,
(3) the three Propositions are so related that, if the first two were true, the third would be true,
the Trio is called a 'Syllogism'; the Genus, of which each of the six Terms is a Species, is called its 'Universe of Discourse', or, more briefly, its 'Univ.'; the first two Propositions are called its 'Premisses', and the third its 'Conclusion'; also the Pair of codivisional Terms in the Premisses are called its 'Eliminands', and the other two its 'Retinends'.
The Conclusion of a Syllogism is said to be 'consequent' from its Premisses: hence it is usual to prefix to it the word "Therefore" (or the Symbol ".'."). pg057 [Note that the 'Eliminands' are so called because they are eliminated, and do not appear in the Conclusion; and that the 'Retinends' are so called because they are retained, and do appear in the Conclusion.
Note also that the question, whether the Conclusion is or is not consequent from the Premisses, is not affected by the actual truth or falsity of any of the Trio, but depends entirely on their relationship to each other.
As a specimen-Syllogism, let us take the Trio
"No x-Things are m-Things; No y-Things are m'-Things. No x-Things are y-Things."
which we may write, as explained at p. 26, thus:—
"No x are m; No y are m'. No x are y".
Here the first and second contain the Pair of codivisional Classes m and m'; the first and third contain the Pair x and x; and the second and third contain the Pair y and y.
Also the three Propositions are (as we shall see hereafter) so related that, if the first two were true, the third would also be true.
Hence the Trio is a Syllogism; the two Propositions, "No x are m" and "No y are m'", are its Premisses; the Proposition "No x are y" is its Conclusion; the Terms m and m' are its Eliminands; and the Terms x and y are its Retinends.
Hence we may write it thus:—
"No x are m; No y are m'. .'. No x are y".
As a second specimen, let us take the Trio
"All cats understand French; Some chickens are cats. Some chickens understand French".
These, put into normal form, are
"All cats are creatures understanding French; Some chickens are cats. Some chickens are creatures understanding French".
Here all the six Terms are Species of the Genus "creatures."
Also the first and second Propositions contain the Pair of codivisional Classes "cats" and "cats"; the first and third contain the Pair "creatures understanding French" and "creatures understanding French"; and the second and third contain the Pair "chickens" and "chickens". pg058 Also the three Propositions are (as we shall see at p. 64) so related that, if the first two were true, the third would be true. (The first two are, as it happens, not strictly true in our planet. But there is nothing to hinder them from being true in some other planet, say Mars or Jupiter—in which case the third would also be true in that planet, and its inhabitants would probably engage chickens as nursery-governesses. They would thus secure a singular contingent privilege, unknown in England, namely, that they would be able, at any time when provisions ran short, to utilise the nursery-governess for the nursery-dinner!)
Hence the Trio is a Syllogism; the Genus "creatures" is its 'Univ.'; the two Propositions, "All cats understand French" and "Some chickens are cats", are its Premisses, the Proposition "Some chickens understand French" is its Conclusion; the Terms "cats" and "cats" are its Eliminands; and the Terms, "creatures understanding French" and "chickens", are its Retinends.
Hence we may write it thus:—
"All cats understand French; Some chickens are cats; .'. Some chickens understand French".]
pg059
CHAPTER II.
PROBLEMS IN SYLLOGISMS.
Sec. 1.
Introductory.
When the Terms of a Proposition are represented by words, it is said to be 'concrete'; when by letters, 'abstract.'
To translate a Proposition from concrete into abstract form, we fix on a Univ., and regard each Term as a Species of it, and we choose a letter to represent its Differentia.
[For example, suppose we wish to translate "Some soldiers are brave" into abstract form. We may take "men" as Univ., and regard "soldiers" and "brave men" as Species of the Genus "men"; and we may choose x to represent the peculiar Attribute (say "military") of "soldiers," and y to represent "brave." Then the Proposition may be written "Some military men are brave men"; i.e. "Some x-men are y-men"; i.e. (omitting "men," as explained at p. 26) "Some x are y."
In practice, we should merely say "Let Univ. be "men", x = soldiers, y = brave", and at once translate "Some soldiers are brave" into "Some x are y."]
The Problems we shall have to solve are of two kinds, viz.
(1) "Given a Pair of Propositions of Relation, which contain between them a pair of codivisional Classes, and which are proposed as Premisses: to ascertain what Conclusion, if any, is consequent from them."
(2) "Given a Trio of Propositions of Relation, of which every two contain a pair of codivisional Classes, and which are proposed as a Syllogism: to ascertain whether the proposed Conclusion is consequent from the proposed Premisses, and, if so, whether it is complete."
These Problems we will discuss separately.
pg060 Sec. 2.
Given a Pair of Propositions of Relation, which contain between them a pair of codivisional Classes, and which are proposed as Premisses: to ascertain what Conclusion, if any, is consequent from them.
The Rules, for doing this, are as follows:—
(1) Determine the 'Universe of Discourse'.
(2) Construct a Dictionary, making m and m (or m and m') represent the pair of codivisional Classes, and x (or x') and y (or y') the other two.
(3) Translate the proposed Premisses into abstract form.
(4) Represent them, together, on a Triliteral Diagram.
(5) Ascertain what Proposition, if any, in terms of x and y, is also represented on it.
(6) Translate this into concrete form.
It is evident that, if the proposed Premisses were true, this other Proposition would also be true. Hence it is a Conclusion consequent from the proposed Premisses.
[Let us work some examples.
(1)
"No son of mine is dishonest; People always treat an honest man with respect".
Taking "men" as Univ., we may write these as follows:—
"No sons of mine are dishonest men; All honest men are men treated with respect".
We can now construct our Dictionary, viz. m = honest; x = sons of mine; y = treated with respect.
(Note that the expression "x = sons of mine" is an abbreviated form of "x = the Differentia of 'sons of mine', when regarded as a Species of 'men'".)
The next thing is to translate the proposed Premisses into abstract form, as follows:—
"No x are m'; All m are y".
pg061 Next, by the process described at p. 50, we represent these on a Triliteral Diagram, thus:—
. -. (O) (O) . - -. (O) - (I) - - (O) . - -. . -.
Next, by the process described at p. 53, we transfer to a Biliteral Diagram all the information we can.
. -. (O) - - . -.
The result we read as "No x are y'" or as "No y' are x," whichever we prefer. So we refer to our Dictionary, to see which will look best; and we choose
"No x are y'",
which, translated into concrete form, is
"No son of mine fails to be treated with respect".
(2)
"All cats understand French; Some chickens are cats".
Taking "creatures" as Univ., we write these as follows:—
"All cats are creatures understanding French; Some chickens are cats".
We can now construct our Dictionary, viz. m = cats; x = understanding French; y = chickens.
The proposed Premisses, translated into abstract form, are
"All m are x; Some y are m".
In order to represent these on a Triliteral Diagram, we break up the first into the two Propositions to which it is equivalent, and thus get the three Propositions
(1) "Some m are x; (2) No m are x'; (3) Some y are m".
The Rule, given at p. 50, would make us take these in the order 2, 1, 3.
This, however, would produce the result
. -. . . (I)(I) (O) (O) . . . -.
pg062 So it would be better to take them in the order 2, 3, 1. Nos. (2) and (3) give us the result here shown; and now we need not trouble about No. (1), as the Proposition "Some m are x" is already represented on the Diagram.
. -. . - -. (I) - - - - (O) (O) . - -. . -.
Transferring our information to a Biliteral Diagram, we get
. -. (I) - - . -.
This result we can read either as "Some x are y" or "Some y are x".
After consulting our Dictionary, we choose
"Some y are x",
which, translated into concrete form, is
"Some chickens understand French."
(3)
"All diligent students are successful; All ignorant students are unsuccessful".
Let Univ. be "students"; m = successful; x = diligent; y = ignorant.
These Premisses, in abstract form, are
"All x are m; All y are m'".
These, broken up, give us the four Propositions
(1) "Some x are m; (2) No x are m'; (3) Some y are m'; (4) No y are m".
which we will take in the order 2, 4, 1, 3.
Representing these on a Triliteral Diagram, we get
. -. (O) (O) . - -. (O) (I) - - - - (O) . - -. (I) . -.
And this information, transferred to a Biliteral Diagram, is
. -. (O) (I) - - (I) . -.
Here we get two Conclusions, viz.
"All x are y'; All y are x'." pg063 And these, translated into concrete form, are
"All diligent students are (not-ignorant, i.e.) learned; All ignorant students are (not-diligent, i.e.) idle". (See p. 4.)
(4)
"Of the prisoners who were put on their trial at the last Assizes, all, against whom the verdict 'guilty' was returned, were sentenced to imprisonment; Some, who were sentenced to imprisonment, were also sentenced to hard labour".
Let Univ. be "the prisoners who were put on their trial at the last Assizes"; m = who were sentenced to imprisonment; x = against whom the verdict 'guilty' was returned; y = who were sentenced to hard labour.
The Premisses, translated into abstract form, are
"All x are m; Some m are y".
Breaking up the first, we get the three
(1) "Some x are m; (2) No x are m'; (3) Some m are y".
Representing these, in the order 2, 1, 3, on a Triliteral Diagram, we get
. -. (O) (O) . - -. (I) - (I) - - . - -. . -.
Here we get no Conclusion at all.
You would very likely have guessed, if you had seen only the Premisses, that the Conclusion would be
"Some, against whom the verdict 'guilty' was returned, were sentenced to hard labour".
But this Conclusion is not even true, with regard to the Assizes I have here invented.
"Not true!" you exclaim. "Then who were they, who were sentenced to imprisonment and were also sentenced to hard labour? They must have had the verdict 'guilty' returned against them, or how could they be sentenced?"
Well, it happened like this, you see. They were three ruffians, who had committed highway-robbery. When they were put on their trial, they pleaded 'guilty'. So no verdict was returned at all; and they were sentenced at once.]
I will now work out, in their briefest form, as models for the Reader to imitate in working examples, the above four concrete Problems. pg064 (1) [see p. 60]
"No son of mine is dishonest; People always treat an honest man with respect."
Univ. "men"; m = honest; x = my sons; y = treated with respect.
. -. . -. "No x are m'; (O) (O) (O) All m are y." . - -. - - (O) - (I) - - . -. (O) . - -. .'. "No x are y'." . -.
i.e. "No son of mine ever fails to be treated with respect."
(2) [see p. 61]
"All cats understand French; Some chickens are cats".
Univ. "creatures"; m = cats; x = understanding French; y = chickens.
. -. . -. "All m are x; (I) Some y are m." . - -. - - (I) - - - - . -. (O) (O) . - -. .'. "Some y are x." . -.
i.e. "Some chickens understand French."
(3) [see p. 62]
"All diligent students are successful; All ignorant students are unsuccessful".
Univ. "students"; m = successful; x = diligent; y = ignorant.
. -. . -. "All x are m; (O) (O) (O) (I) All y are m'." . - -. - - (O) (I) (I) - - - - . -. (O) . - -. .'. "All x are y'; (I) All y are x'." . -.
i.e. "All diligent students are learned; and all ignorant students are idle". pg065 (4) [see p. 63]
"Of the prisoners who were put on their trial at the last Assizes, all, against whom the verdict 'guilty' was returned, were sentenced to imprisonment;
Some, who were sentenced to imprisonment, were also sentenced to hard labour".
Univ. "prisoners who were put on their trial at the last Assizes", m = sentenced to imprisonment; x = against whom the verdict 'guilty' was returned; y = sentenced to hard labour.
. -. "All x are m; (O) (O) Some m are y." . - -. (I) - (I) - - . - -. There is no Conclusion. . -.
[Review Tables VII, VIII (pp. 48, 49). Work Examples Sec. 1, 17-21 (p. 97); Sec. 4, 1-6 (p. 100); Sec. 5, 1-6 (p. 101).]
pg066 Sec. 3.
Given a Trio of Propositions of Relation, of which every two contain a Pair of codivisional Classes, and which are proposed as a Syllogism; to ascertain whether the proposed Conclusion is consequent from the proposed Premisses, and, if so, whether it is complete.
The Rules, for doing this, are as follows:—
(1) Take the proposed Premisses, and ascertain, by the process described at p. 60, what Conclusion, if any, is consequent from them.
(2) If there be no Conclusion, say so.
(3) If there be a Conclusion, compare it with the proposed Conclusion, and pronounce accordingly.
I will now work out, in their briefest form, as models for the Reader to imitate in working examples, six Problems.
(1)
"All soldiers are strong; All soldiers are brave. Some strong men are brave."
Univ. "men"; m = soldiers; x = strong; y = brave. pg067 . -. . -. "All m are x; (I) All m are y. . - -. - - Some x are y." (I) (O) - - - - . -. (O) (O) . - -. .'. "Some x are y." . -.
Hence proposed Conclusion is right.
(2)
"I admire these pictures; When I admire anything I wish to examine it thoroughly. I wish to examine some of these pictures thoroughly."
Univ. "things"; m = admired by me; x = these pictures; y = things which I wish to examine thoroughly.
. -. . -. "All x are m; (O) (O) (I) (O) All m are y. . - -. - - Some x are y." (I) (O) - - - - . -. (O) . - -. .'. "All x are y." . -.
Hence proposed Conclusion is incomplete, the complete one being "I wish to examine all these pictures thoroughly".
(3)
"None but the brave deserve the fair; Some braggarts are cowards. Some braggarts do not deserve the fair."
Univ. "persons"; m = brave; x = deserving of the fair; y = braggarts.
. -. . -. "No m' are x; (O) (O) Some y are m'. . - -. - - Some y are x'." (I) - - - - . -. . - -. .'. "Some y are x'." (I) . -.
Hence proposed Conclusion is right. pg068 (4)
"All soldiers can march; Some babies are not soldiers. Some babies cannot march".
Univ. "persons"; m = soldiers; x = able to march; y = babies.
. -. "All m are x; Some y are m'. . - -. Some y are x'." (I) (I) - - - (O) (O) . - -. There is no Conclusion. . -.
(5)
"All selfish men are unpopular; All obliging men are popular. All obliging men are unselfish".
Univ. "men"; m = popular; x = selfish; y = obliging.
. -. . -. "All x are m'; (O) (I) (O) (I) All y are m. . - -. - - All y are x'." (O) (O) (I) - - - - . -. (I) . - -. .'. "All x are y'; (O) All y are x'." . -.
Hence proposed Conclusion is incomplete, the complete one containing, in addition, "All selfish men are disobliging".
(6)
"No one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running;
This party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station.
This party of tourists need not run."
Univ. "persons meaning to go by the train, and unable to get a conveyance"; m = having enough time to walk to the station; x = needing to run; y = these tourists. pg069 . -. "No m' are x'; (O) All y are m. . - -. All y are x'." - (I) - - . - -. There is no (O) (O) Conclusion. . -.
[Here is another opportunity, gentle Reader, for playing a trick on your innocent friend. Put the proposed Syllogism before him, and ask him what he thinks of the Conclusion.
He will reply "Why, it's perfectly correct, of course! And if your precious Logic-book tells you it isn't, don't believe it! You don't mean to tell me those tourists need to run? If I were one of them, and knew the Premisses to be true, I should be quite clear that I needn't run—and I should walk!"
And you will reply "But suppose there was a mad bull behind you?"
And then your innocent friend will say "Hum! Ha! I must think that over a bit!"
You may then explain to him, as a convenient test of the soundness of a Syllogism, that, if circumstances can be invented which, without interfering with the truth of the Premisses, would make the Conclusion false, the Syllogism must be unsound.]
[Review Tables V-VIII (pp. 46-49). Work Examples Sec. 4, 7-12 (p. 100); Sec. 5, 7-12 (p. 101); Sec. 6, 1-10 (p. 106); Sec. 7, 1-6 (pp. 107, 108).]
pg070
BOOK VI.
THE METHOD OF SUBSCRIPTS.
CHAPTER I.
INTRODUCTORY.
Let us agree that "x{1}" shall mean "Some existing Things have the Attribute x", i.e. (more briefly) "Some x exist"; also that "xy{1}" shall mean "Some xy exist", and so on. Such a Proposition may be called an 'Entity.'
[Note that, when there are two letters in the expression, it does not in the least matter which stands first: "xy{1}" and "yx{1}" mean exactly the same.]
Also that "x{0}" shall mean "No existing Things have the Attribute x", i.e. (more briefly) "No x exist"; also that "xy{0}" shall mean "No xy exist", and so on. Such a Proposition may be called a 'Nullity'.
Also that "+" shall mean "and".
[Thus "ab{1} + cd{0}" means "Some ab exist and no cd exist".]
Also that ">" shall mean "would, if true, prove".
[Thus, "x{0} > xy{0}" means "The Proposition 'No x exist' would, if true, prove the Proposition 'No xy exist'".]
When two Letters are both of them accented, or both not accented, they are said to have 'Like Signs', or to be 'Like': when one is accented, and the other not, they are said to have 'Unlike Signs', or to be 'Unlike'.
pg071
CHAPTER II.
REPRESENTATION OF PROPOSITIONS OF RELATION.
Let us take, first, the Proposition "Some x are y".
This, we know, is equivalent to the Proposition of Existence "Some xy exist". (See p. 31.) Hence it may be represented by the expression "xy_{1}".
The Converse Proposition "Some y are x" may of course be represented by the _same_ expression, viz. "xy_{1}".
Similarly we may represent the three similar Pairs of Converse Propositions, viz.—
"Some x are y'" = "Some y' are x", "Some x' are y" = "Some y are x'", "Some x' are y'" = "Some y' are x'".
Let us take, next, the Proposition "No x are y".
This, we know, is equivalent to the Proposition of Existence "No xy exist". (See p. 33.) Hence it may be represented by the expression "xy_{0}".
The Converse Proposition "No y are x" may of course be represented by the _same_ expression, viz. "xy_{0}".
Similarly we may represent the three similar Pairs of Converse Propositions, viz.—
"No x are y'" = "No y' are x", "No x' are y" = "No y are x'", "No x' are y'" = "No y' are x'". pg072 Let us take, next, the Proposition "All x are y".
Now it is evident that the Double Proposition of Existence "Some x exist and no xy' exist" tells us that some x-Things exist, but that none of them have the Attribute y': that is, it tells us that all of them have the Attribute y: that is, it tells us that "All x are y".
Also it is evident that the expression "x{1} + xy'{0}" represents this Double Proposition.
Hence it also represents the Proposition "All x are y".
[The Reader will perhaps be puzzled by the statement that the Proposition "All x are y" is equivalent to the Double Proposition "Some x exist and no xy' exist," remembering that it was stated, at p. 33, to be equivalent to the Double Proposition "Some x are y and no x are y'" (i.e. "Some xy exist and no xy' exist"). The explanation is that the Proposition "Some xy exist" contains superfluous information. "Some x exist" is enough for our purpose.]
This expression may be written in a shorter form, viz. "x{1}y'{0}", since each Subscript takes effect back to the beginning of the expression.
Similarly we may represent the seven similar Propositions "All x are y'", "All x' are y", "All x' are y'", "All y are x", "All y are x'", "All y' are x", and "All y' are x'".
[The Reader should make out all these for himself.]
It will be convenient to remember that, in translating a Proposition, beginning with "All", from abstract form into subscript form, or vice versa, the Predicate changes sign (that is, changes from positive to negative, or else from negative to positive).
[Thus, the Proposition "All y are x'" becomes "y{1}x{0}", where the Predicate changes from x' to x.
Again, the expression "x'{1}y'{0}" becomes "All x' are y", where the Predicate changes for y' to y.]
pg073
CHAPTER III.
SYLLOGISMS.
Sec. 1.
Representation of Syllogisms.
We already know how to represent each of the three Propositions of a Syllogism in subscript form. When that is done, all we need, besides, is to write the three expressions in a row, with "+" between the Premisses, and ">" before the Conclusion.
[Thus the Syllogism
"No x are m'; All m are y. .'. No x are y'."
may be represented thus:—
xm'{0} + m{1}y'{0} > xy'{0}
When a Proposition has to be translated from concrete form into subscript form, the Reader will find it convenient, just at first, to translate it into abstract form, and thence into subscript form. But, after a little practice, he will find it quite easy to go straight from concrete form to subscript form.]
pg074 Sec. 2.
Formulae for solving Problems in Syllogisms.
When once we have found, by Diagrams, the Conclusion to a given Pair of Premisses, and have represented the Syllogism in subscript form, we have a Formula, by which we can at once find, without having to use Diagrams again, the Conclusion to any other Pair of Premisses having the same subscript forms.
[Thus, the expression
xm_{0} + ym'_{0} > xy_{0}
is a Formula, by which we can find the Conclusion to any Pair of Premisses whose subscript forms are
xm{0} + ym'{0}
For example, suppose we had the Pair of Propositions
"No gluttons are healthy; No unhealthy men are strong".
proposed as Premisses. Taking "men" as our 'Universe', and making m = healthy; x = gluttons; y = strong; we might translate the Pair into abstract form, thus:—
"No x are m; No m' are y".
These, in subscript form, would be
xm{0} + m'y{0}
which are identical with those in our Formula. Hence we at once know the Conclusion to be
xy_{0}
that is, in abstract form,
"No x are y";
that is, in concrete form,
"No gluttons are strong".]
I shall now take three different forms of Pairs of Premisses, and work out their Conclusions, once for all, by Diagrams; and thus obtain some useful Formulae. I shall call them "Fig. I", "Fig. II", and "Fig. III". pg075 Fig. I.
This includes any Pair of Premisses which are both of them Nullities, and which contain Unlike Eliminands.
The simplest case is
. -. . -. xm_{0} + ym'_{0} (O) (O) . - -. - - (O) (O) - - - - . -. . - -. .'. xy_{0} (O) . -.
In this case we see that the Conclusion is a Nullity, and that the Retinends have kept their Signs.
And we should find this Rule to hold good with any Pair of Premisses which fulfil the given conditions.
[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as
m{1}x{0} + ym'{0} (which > xy{0}) xm'{0} + m{1}y{0} (which > xy{0}) x'm{0} + ym'{0} (which > x'y{0}) m'{1}x'{0} + m{1}y'{0} (which > x'y'{0}).]
If either Retinend is asserted in the Premisses to exist, of course it may be so asserted in the Conclusion.
Hence we get two Variants of Fig. I, viz.
(a) where one Retinend is so asserted;
(b) where both are so asserted.
[The Reader had better work out, on Diagrams, examples of these two Variants, such as
m{1}x{0} + y{1}m'{0} (which proves y{1}x{0}) x{1}m'{0} + m{1}y{0} (which proves x{1}y{0}) x'{1}m{0} + y{1}m'{0} (which proves x'{1}y{0} + y{1}x'{0}).]
The Formula, to be remembered, is
xm_{0} + ym'_{0} > xy_{0}
with the following two Rules:—
(1) Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs. pg076 (2) A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion.
[Note that Rule (1) is merely the Formula expressed in words.]
Fig. II.
This includes any Pair of Premisses, of which one is a Nullity and the other an Entity, and which contain Like Eliminands.
The simplest case is
xm{0} + ym{1}
. -. . -. . - -. - - (O) (O) (I) - - - - . -. (I) . - -. .'. x'y_{1} . -.
In this case we see that the Conclusion is an Entity, and that the Nullity-Retinend has changed its Sign.
And we should find this Rule to hold good with any Pair of Premisses which fulfil the given conditions.
[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as
x'm_{0} + ym_{1} (which > xy_{1}) x_{1}m'_{0} + y'm'_{1} (which > x'y'_{1}) m_{1}x_{0} + y'm_{1} (which > x'y'_{1}).]
The Formula, to be remembered, is,
xm_{0} + ym_{1} > x'y_{1}
with the following Rule:—
A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign.
[Note that this Rule is merely the Formula expressed in words.]
pg077 Fig. III.
This includes any Pair of Premisses which are both of them Nullities, and which contain Like Eliminands asserted to exist.
The simplest case is
xm_{0} + ym_{0} + m_{1}
[Note that "m_{1}" is here stated _separately_, because it does not matter in which of the two Premisses it occurs: so that this includes the _three_ forms "m_{1}x_{0} + ym_{0}", "xm_{0} + m_{1}y_{0}", and "m_{1}x_{0} + m_{1}y_{0}".]
. -. . -. . - -. - - (O) (O) (I) - - - - . -. (O) (I) . - -. .'. x'y'_{1} . -.
In this case we see that the Conclusion is an Entity, and that both Retinends have changed their Signs.
And we should find this Rule to hold good with any Pair of Premisses which fulfil the given conditions.
[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as
x'm_{0} + m_{1}y_{0} (which > xy'_{1}) m'_{1}x_{0} + m'y'_{0} (which > x'y_{1}) m_{1}x'_{0} + m_{1}y'_{0} (which > xy_{1}).]
The Formula, to be remembered, is
xm{0} + ym{0} + m{1} > x'y'{1}
with the following Rule (which is merely the Formula expressed in words):—
Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.
* * * * *
In order to help the Reader to remember the peculiarities and Formulae of these three Figures, I will put them all together in one Table. pg078 TABLE IX. Fig. I. xm{0} + ym'{0} > xy{0} Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs. A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion. Fig. II. xm{0} + ym{1} > x'y{1} A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign. Fig. III. xm{0} + ym{0} + m{1} > x'y'{1} Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.
I will now work out, by these Formulae, as models for the Reader to imitate, some Problems in Syllogisms which have been already worked, by Diagrams, in Book V., Chap. II.
(1) [see p. 64]
"No son of mine is dishonest; People always treat an honest man with respect."
Univ. "men"; m = honest; x = my sons; y = treated with respect.
xm'{0} + m{1}y'{0} > xy'{0} [Fig. I.
i.e. "No son of mine ever fails to be treated with respect." pg079 (2) [see p. 64]
"All cats understand French; Some chickens are cats."
Univ. "creatures"; m = cats; x = understanding French; y = chickens.
m{1}x'{0} + ym{1} > xy{1} [Fig. II.
i.e. "Some chickens understand French."
(3) [see p. 64]
"All diligent students are successful; All ignorant students are unsuccessful."
Univ. "students"; m = successful; x = diligent; y = ignorant.
x{1}m'{0} + y{1}m{0} > x{1}y{0} + y{1}x{0} [Fig. I (b).
i.e. "All diligent students are learned; and all ignorant students are idle."
(4) [see p. 66]
"All soldiers are strong; All soldiers are brave. Some strong men are brave."
Univ. "men"; m = soldiers; x = strong; y = brave.
m_{1}x'_{0} + m_{1}y'_{0} > xy_{1} [Fig. III.
Hence proposed Conclusion is right.
(5) [see p. 67]
"I admire these pictures; When I admire anything, I wish to examine it thoroughly. I wish to examine some of these pictures thoroughly."
Univ. "things"; m = admired by me; x = these; y = things which I wish to examine thoroughly.
x{1}m'{0} + m{1}y'{0} > x{1}y'{0} [Fig. I (a).
Hence proposed Conclusion, xy_{1}, is _incomplete_, the _complete_ one being "I wish to examine _all_ these pictures thoroughly." pg080 (6) [see p. 67]
"None but the brave deserve the fair; Some braggarts are cowards. Some braggarts do not deserve the fair."
Univ. "persons"; m = brave; x = deserving of the fair; y = braggarts.
m'x_{0} + ym'_{1} > x'y_{1} [Fig. II.
Hence proposed Conclusion is right.
(7) [see p. 69]
"No one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running;
This party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station.
This party of tourists need not run."
Univ. "persons meaning to go by the train, and unable to get a conveyance"; m = having enough time to walk to the station; x = needing to run; y = these tourists.
m'x'_{0} + y_{1}m'_{0} do not come under any of the three Figures. Hence it is necessary to return to the Method of Diagrams, as shown at p. 69.
Hence there is no Conclusion.
[Work Examples Sec. 4, 12-20 (p. 100); Sec. 5, 13-24 (pp. 101, 102); Sec. 6, 1-6 (p. 106); Sec. 7, 1-3 (pp. 107, 108). Also read Note (A), at p. 164.]
pg081 Sec. 3.
Fallacies.
Any argument which deceives us, by seeming to prove what it does not really prove, may be called a 'Fallacy' (derived from the Latin verb fallo "I deceive"): but the particular kind, to be now discussed, consists of a Pair of Propositions, which are proposed as the Premisses of a Syllogism, but yield no Conclusion.
When each of the proposed Premisses is a Proposition in I, or E, or A, (the only kinds with which we are now concerned,) the Fallacy may be detected by the 'Method of Diagrams,' by simply setting them out on a Triliteral Diagram, and observing that they yield no information which can be transferred to the Biliteral Diagram.
But suppose we were working by the 'Method of Subscripts,' and had to deal with a Pair of proposed Premisses, which happened to be a 'Fallacy,' how could we be certain that they would not yield any Conclusion?
Our best plan is, I think, to deal with Fallacies in the same was as we have already dealt with Syllogisms: that is, to take certain forms of Pairs of Propositions, and to work them out, once for all, on the Triliteral Diagram, and ascertain that they yield no Conclusion; and then to record them, for future use, as Formulae for Fallacies, just as we have already recorded our three Formulae for Syllogisms. pg082 Now, if we were to record the two Sets of Formulae in the same shape, viz. by the Method of Subscripts, there would be considerable risk of confusing the two kinds. Hence, in order to keep them distinct, I propose to record the Formulae for Fallacies in words, and to call them "Forms" instead of "Formulae."
Let us now proceed to find, by the Method of Diagrams, three "Forms of Fallacies," which we will then put on record for future use. They are as follows:—
(1) Fallacy of Like Eliminands not asserted to exist. (2) Fallacy of Unlike Eliminands with an Entity-Premiss. (3) Fallacy of two Entity-Premisses.
These shall be discussed separately, and it will be seen that each fails to yield a Conclusion.
(1) Fallacy of Like Eliminands not asserted to exist.
It is evident that neither of the given Propositions can be an Entity, since that kind asserts the existence of both of its Terms (see p. 20). Hence they must both be Nullities.
Hence the given Pair may be represented by (xm{0} + ym{0}), with or without x{1}, y{1}.
These, set out on Triliteral Diagrams, are
xm_{0} + ym_{0} x_{1}m_{0} + ym_{0} . -. . -. (I) . - -. . - -. (O) (O) (O) (O) - - - - - - - - (O) (O) . - -. . - -. . -. . -.
xm_{0} + y_{1}m_{0} x_{1}m_{0} + y_{1}m_{0} . -. . -. (I) . - -. . - -. (O) (O) (O) (O) (I) - - - (I) - - - (O) (O) . - -. . - -. . -. . -. pg083 (2) _Fallacy of Unlike Eliminands with an Entity-Premiss._
Here the given Pair may be represented by (xm{0} + ym'{1}) with or without x{1} or m{1}.
These, set out on Triliteral Diagrams, are
xm{0} + ym'{1} x{1}m{0} + ym'{1} m{1}x{0} + ym'{1} . -. . -. . -. (I) . - -. . - -. . - -. (O) (O) (O) (O) (O) (O) (I) - - - (I) - - - (I) - - - (I) . - -. . - -. . - -. . -. . -. . -.
(3) Fallacy of two Entity-Premisses.
Here the given Pair may be represented by either (xm{1} + ym{1}) or (xm{1} + ym'{1}).
These, set out on Triliteral Diagrams, are
xm{1} + ym{1} xm{1} + ym'{1} . -. . -. . - -. . - -. (I) (I) - (I) - - (I) - - - . - -. . - -. . -. . -.
pg084 Sec. 4.
Method of proceeding with a given Pair of Propositions.
Let us suppose that we have before us a Pair of Propositions of Relation, which contain between them a Pair of codivisional Classes, and that we wish to ascertain what Conclusion, if any, is consequent from them. We translate them, if necessary, into subscript-form, and then proceed as follows:—
(1) We examine their Subscripts, in order to see whether they are
(a) a Pair of Nullities; or (b) a Nullity and an Entity; or (c) a Pair of Entities.
(2) If they are a Pair of Nullities, we examine their Eliminands, in order to see whether they are Unlike or Like.
If their Eliminands are Unlike, it is a case of Fig. I. We then examine their Retinends, to see whether one or both of them are asserted to exist. If one Retinend is so asserted, it is a case of Fig. I (a); if both, it is a case of Fig. I (b).
If their Eliminands are Like, we examine them, in order to see whether either of them is asserted to exist. If so, it is a case of Fig. III.; if not, it is a case of "Fallacy of Like Eliminands not asserted to exist."
(3) If they are a Nullity and an Entity, we examine their Eliminands, in order to see whether they are Like or Unlike.
If their Eliminands are Like, it is a case of Fig. II.; if Unlike, it is a case of "Fallacy of Unlike Eliminands with an Entity-Premiss."
(4) If they are a Pair of Entities, it is a case of "Fallacy of two Entity-Premisses."
[Work Examples Sec. 4, 1-11 (p. 100); Sec. 5, 1-12 (p. 101); Sec. 6, 7-12 (p. 106); Sec. 7, 7-12 (p. 108).]
pg085
BOOK VII.
SORITESES.
CHAPTER I.
INTRODUCTORY.
When a Set of three or more Biliteral Propositions are such that all their Terms are Species of the same Genus, and are also so related that two of them, taken together, yield a Conclusion, which, taken with another of them, yields another Conclusion, and so on, until all have been taken, it is evident that, if the original Set were true, the last Conclusion would also be true.
Such a Set, with the last Conclusion tacked on, is called a 'Sorites'; the original Set of Propositions is called its 'Premisses'; each of the intermediate Conclusions is called a 'Partial Conclusion' of the Sorites; the last Conclusion is called its 'Complete Conclusion,' or, more briefly, its 'Conclusion'; the Genus, of which all the Terms are Species, is called its 'Universe of Discourse', or, more briefly, its 'Univ.'; the Terms, used as Eliminands in the Syllogisms, are called its 'Eliminands'; and the two Terms, which are retained, and therefore appear in the Conclusion, are called its 'Retinends'.
[Note that each Partial Conclusion contains one or two Eliminands; but that the Complete Conclusion contains Retinends only.]
The Conclusion is said to be 'consequent' from the Premisses; for which reason it is usual to prefix to it the word "Therefore" (or the symbol ".'.").
[Note that the question, whether the Conclusion is or is not consequent from the Premisses, is not affected by the actual truth or falsity of any one of the Propositions which make up the Sorites, by depends entirely on their relationship to one another. pg086 As a specimen-Sorites, let us take the following Set of 5 Propositions:—
(1) "No a are b'; (2) All b are c; (3) All c are d; (4) No e' are a'; (5) All h are e'".
Here the first and second, taken together, yield "No a are c'".
This, taken along with the third, yields "No a are d'".
This, taken along with the fourth, yields "No d' are e'".
And this, taken along with the fifth, yields "All h are d".
Hence, if the original Set were true, this would also be true.
Hence the original Set, with this tacked on, is a Sorites; the original Set is its Premisses; the Proposition "All h are d" is its Conclusion; the Terms a, b, c, e are its Eliminands; and the Terms d and h are its Retinends.
Hence we may write the whole Sorites thus:—
"No a are b'; All b are c; All c are d; No e' are a'; All h are e'. .'. All h are d".
In the above Sorites, the 3 Partial Conclusions are the Positions "No a are e'", "No a are d'", "No d' are e'"; but, if the Premisses were arranged in other ways, other Partial Conclusions might be obtained. Thus, the order 41523 yields the Partial Conclusions "No c' are b'", "All h are b", "All h are c". There are altogether nine Partial Conclusions to this Sorites, which the Reader will find it an interesting task to make out for himself.]
pg087
CHAPTER II.
PROBLEMS IN SORITESES.
Sec. 1.
Introductory.
The Problems we shall have to solve are of the following form:—
"Given three or more Propositions of Relation, which are proposed as Premisses: to ascertain what Conclusion, if any, is consequent from them."
We will limit ourselves, at present, to Problems which can be worked by the Formulae of Fig. I. (See p. 75.) Those, that require other Formulae, are rather too hard for beginners.
Such Problems may be solved by either of two Methods, viz.
(1) The Method of Separate Syllogisms; (2) The Method of Underscoring.
These shall be discussed separately.
pg088 Sec. 2.
Solution by Method of Separate Syllogisms.
The Rules, for doing this, are as follows:—
(1) Name the 'Universe of Discourse'.
(2) Construct a Dictionary, making a, b, c, &c. represent the Terms.
(3) Put the Proposed Premisses into subscript form.
(4) Select two which, containing between them a pair of codivisional Classes, can be used as the Premisses of a Syllogism.
(5) Find their Conclusion by Formula.
(6) Find a third Premiss which, along with this Conclusion, can be used as the Premisses of a second Syllogism.
(7) Find a second Conclusion by Formula.
(8) Proceed thus, until all the proposed Premisses have been used.
(9) Put the last Conclusion, which is the Complete Conclusion of the Sorites, into concrete form.
[As an example of this process, let us take, as the proposed Set of Premisses,
(1) "All the policemen on this beat sup with our cook; (2) No man with long hair can fail to be a poet; (3) Amos Judd has never been in prison; (4) Our cook's 'cousins' all love cold mutton; (5) None but policemen on this beat are poets; (6) None but her 'cousins' ever sup with our cook; (7) Men with short hair have all been in prison."
Univ. "men"; a = Amos Judd; b = cousins of our cook; c = having been in prison; d = long-haired; e = loving cold mutton; h = poets; k = policemen on this beat; l = supping with our cook pg089 We now have to put the proposed Premisses into subscript form. Let us begin by putting them into abstract form. The result is
(1) "All k are l; (2) No d are h'; (3) All a are c'; (4) All b are e; (5) No k' are h; (6) No b' are l; (7) All d' are c."
And it is now easy to put them into subscript form, as follows:—
(1) k_{1}l'_{0} (2) dh'_{0} (3) a_{1}c_{0} (4) b_{1}e'_{0} (5) k'h_{0} (6) b'l_{0} (7) d'_{1}c'_{0}
We now have to find a pair of Premisses which will yield a Conclusion. Let us begin with No. (1), and look down the list, till we come to one which we can take along with it, so as to form Premisses belonging to Fig. I. We find that No. (5) will do, since we can take k as our Eliminand. So our first syllogism is
(1) k{1}l'{0} (5) k'h{0} .'. l'h{0} ... (8)
We must now begin again with l'h_{0} and find a Premiss to go along with it. We find that No. (2) will do, h being our Eliminand. So our next Syllogism is
(8) l'h_{0} (2) dh'_{0} .'. l'd_{0} ... (9)
We have now used up Nos. (1), (5), and (2), and must search among the others for a partner for l'd_{0}. We find that No. (6) will do. So we write
(9) l'd_{0} (6) b'l_{0} .'. db'_{0} ... (10)
Now what can we take along with db'_{0}? No. (4) will do.
(10) db'{0} (4) b{1}e'{0} .'. de'{0} ... (11) pg090 Along with this we may take No. (7).
(11) de'{0} (7) d'{1}c'{0} .'. c'e'{0} ... (12)
And along with this we may take No. (3).
(12) c'e'_{0} (3) a_{1}c_{0} .'. a_{1}e'_{0}
This Complete Conclusion, translated into abstract form, is
"All a are e";
and this, translated into concrete form, is
"Amos Judd loves cold mutton."
In actually working this Problem, the above explanations would, of course, be omitted, and all, that would appear on paper, would be as follows:—
(1) k_{1}l'_{0} (2) dh'_{0} (3) a_{1}c_{0} (4) b_{1}e'_{0} (5) k'h_{0} (6) b'l_{0} (7) d'_{1}c'_{0}
(1) k{1}l'{0} (5) k'h{0} .'. l'h{0} ... (8)
(8) l'h_{0} (2) dh'_{0} .'. l'd_{0} ... (9)
(9) l'd_{0} (6) b'l_{0} .'. db'_{0} ... (10)
(10) db'{0} (4) b{1}e'{0} .'. de'{0} ... (11)
(11) de'{0} (7) d'{1}c'{0} .'. c'e'{0} ... (12)
(12) c'e'_{0} (3) a_{1}c_{0} .'. a_{1}e'_{0}
Note that, in working a Sorites by this Process, we may begin with any Premiss we choose.]
pg091 Sec. 3.
Solution by Method of Underscoring.
Consider the Pair of Premisses
xm{0} + ym'{0}
which yield the Conclusion xy_{0}
We see that, in order to get this Conclusion, we must eliminate m and m', and write x and y together in one expression.
Now, if we agree to mark m and m' as eliminated, and to read the two expressions together, as if they were written in one, the two Premisses will then exactly represent the Conclusion, and we need not write it out separately.
Let us agree to mark the eliminated letters by underscoring them, putting a single score under the first, and a double one under the second.
The two Premisses now become
xm{0} + ym'{0} - =
which we read as "xy_{0}".
In copying out the Premisses for underscoring, it will be convenient to omit all subscripts. As to the "0's" we may always suppose them written, and, as to the "1's", we are not concerned to know which Terms are asserted to exist, except those which appear in the Complete Conclusion; and for them it will be easy enough to refer to the original list. pg092 [I will now go through the process of solving, by this method, the example worked in Sec. 2.
The Data are
1 2 3 4 k_{1}l'_{0} + dh'_{0} + a_{1}c_{0} + b_{1}e'_{0} +
5 6 7 k'h{0} + b'l{0} + d'{1}c'{0}
The Reader should take a piece of paper, and write out this solution for himself. The first line will consist of the above Data; the second must be composed, bit by bit, according to the following directions.
We begin by writing down the first Premiss, with its numeral over it, but omitting the subscripts.
We have now to find a Premiss which can be combined with this, i.e., a Premiss containing either k' or l. The first we find is No. 5; and this we tack on, with a +.
To get the Conclusion from these, k and k' must be eliminated, and what remains must be taken as one expression. So we underscore them, putting a single score under k, and a double one under k'. The result we read as l'h.
We must now find a Premiss containing either l or h'. Looking along the row, we fix on No. 2, and tack it on.
Now these 3 Nullities are really equivalent to (l'h + dh'), in which h and h' must be eliminated, and what remains taken as one expression. So we underscore them. The result reads as l'd.
We now want a Premiss containing l or d'. No. 6 will do.
These 4 Nullities are really equivalent to (l'd + b'l). So we underscore l' and l. The result reads as db'.
We now want a Premiss containing d' or b. No. 4 will do.
Here we underscore b' and b. The result reads as de'.
We now want a Premiss containing d' or e. No. 7 will do.
Here we underscore d and d'. The result reads as c'e'.
We now want a Premiss containing c or e. No. 3 will do—in fact must do, as it is the only one left.
Here we underscore c' and c; and, as the whole thing now reads as e'a, we tack on e'a_{0} as the _Conclusion_, with a >.
We now look along the row of Data, to see whether e' or a has been given as existent. We find that a has been so given in No. 3. So we add this fact to the Conclusion, which now stands as > e'a{0} + a{1}, i.e. > a{1}e'{0}; i.e. "All a are e."
If the Reader has faithfully obeyed the above directions, his written solution will now stand as follows:—
1 2 3 4 k_{1}l'_{0} + dh'_{0} + a_{1}c_{0} + b_{1}e'_{0} +
5 6 7 k'h{0} + b'l{0} + d'{1}c'{0}
1 5 2 6 4 7 3 kl' + k'h + dh' + b'l + be' + d'c' + ac — = - -= - = = = - =
> e'a{0} + a{1}
i.e. > a{1}e'{0};
i.e. "All a are e." pg093 The Reader should now take a second piece of paper, and copy the Data only, and try to work out the solution for himself, beginning with some other Premiss.
If he fails to bring out the Conclusion a{1}e'{0}, I would advise him to take a third piece of paper, and begin again!]
I will now work out, in its briefest form, a Sorites of 5 Premisses, to serve as a model for the Reader to imitate in working examples.
(1) "I greatly value everything that John gives me; (2) Nothing but this bone will satisfy my dog; (3) I take particular care of everything that I greatly value; (4) This bone was a present from John; (5) The things, of which I take particular care, are things I do not give to my dog".
Univ. "things"; a = given by John to me; b = given by me to my dog; c = greatly valued by me; d = satisfactory to my dog; e = taken particular care of by me; h = this bone.
1 2 3 4 5 a_{1}c'_{0} + h'd_{0} + c_{1}e'_{0} + h_{1}a'_{0} + e_{1}b_{0}
1 3 4 2 5 ac' + ce' + ha' + h'd + eb > db_{0} — =- -= = =
i.e. "Nothing, that I give my dog, satisfies him," or, "My dog is not satisfied with anything that I give him!"
[Note that, in working a Sorites by this process, we may begin with any Premiss we choose. For instance, we might begin with No. 5, and the result would then be
5 3 1 4 2 eb + ce' + ac' + ha' + h'd > bd_{0}] - -= -= -= =
[Work Examples Sec. 4, 25-30 (p. 100); Sec. 5, 25-30 (p. 102); Sec. 6, 13-15 (p. 106); Sec. 7, 13-15 (p. 108); Sec. 8, 1-4, 13, 14, 19, 24 (pp. 110, 111); Sec. 9, 1-4, 26, 27, 40, 48 (pp. 112, 116, 119, 121).]
pg094 The Reader, who has successfully grappled with all the Examples hitherto set, and who thirsts, like Alexander the Great, for "more worlds to conquer," may employ his spare energies on the following 17 Examination-Papers. He is recommended not to attempt more than one Paper on any one day. The answers to the questions about words and phrases may be found by referring to the Index at p. 197.
I. Sec. 4, 31 (p. 100); Sec. 5, 31-34 (p. 102); Sec. 6, 16, 17 (p. 106); Sec. 7, 16 (p. 108); Sec. 8, 5, 6 (p. 110); Sec. 9, 5, 22, 42 (pp. 112, 115, 119). What is 'Classification'? And what is a 'Class'?
II. Sec. 4, 32 (p. 100); Sec. 5, 35-38 (pp. 102, 103); Sec. 6, 18 (p. 107); Sec. 7, 17, 18 (p. 108); Sec. 8, 7, 8 (p. 110); Sec. 9, 6, 23, 43 (pp. 112, 115, 119). What are 'Genus', 'Species', and 'Differentia'?
III. Sec. 4, 33 (p. 100); Sec. 5, 39-42 (p. 103); Sec. 6, 19, 20 (p. 107); Sec. 7, 19 (p. 109); Sec. 8, 9, 10 (p. 111); Sec. 9, 7, 24, 44 (pp. 113, 116, 120). What are 'Real' and 'Imaginary' Classes?
IV. Sec. 4, 34 (p. 100); Sec. 5, 43-46 (p. 103); Sec. 6, 21 (p. 107); Sec. 7, 20, 21 (p. 109); Sec. 8, 11, 12 (p. 111); Sec. 9, 8, 25, 45 (pp. 113, 116, 120). What is 'Division'? When are Classes said to be 'Codivisional'?
V. Sec. 4, 35 (p. 100); Sec. 5, 47-50 (p. 103); Sec. 6, 22, 23 (p. 107); Sec. 7, 22 (p. 109); Sec. 8, 15, 16 (p. 111); Sec. 9, 9, 28, 46 (pp. 113, 116, 120). What is 'Dichotomy'? What arbitrary rule does it sometimes require? pg095 VI. Sec. 4, 36 (p. 100); Sec. 5, 51-54 (p. 103); Sec. 6, 24 (p. 107); Sec. 7, 23, 24 (p. 109); Sec. 8, 17 (p. 111); Sec. 9, 10, 29, 47 (pp. 113, 117, 120). What is a 'Definition'?
VII. Sec. 4, 37 (p. 100); Sec. 5, 55-58 (pp. 103, 104); Sec. 6, 25, 26 (p. 107); Sec. 7, 25 (p. 109); Sec. 8, 18 (p. 111); Sec. 9, 11, 30, 49 (pp. 113, 117, 121). What are the 'Subject' and the 'Predicate' of a Proposition? What is its 'Normal' form?
VIII. Sec. 4, 38 (p. 100); Sec. 5, 59-62 (p. 104); Sec. 6, 27 (p. 107); Sec. 7, 26, 27 (p. 109); Sec. 8, 20 (p. 111); Sec. 9, 12, 31, 50 (pp. 113, 117, 121). What is a Proposition 'in I'? 'In E'? And 'in A'?
IX. Sec. 4, 39 (p. 100); Sec. 5, 63-66 (p. 104); Sec. 6, 28, 29 (p. 107); Sec. 7, 28 (p. 109); Sec. 8, 21 (p. 111); Sec. 9, 13, 32, 51 (pp. 114, 117, 121). What is the 'Normal' form of a Proposition of Existence?
X. Sec. 4, 40 (p. 100); Sec. 5, 67-70 (p. 104); Sec. 6, 30 (p. 107); Sec. 7, 29, 30 (p. 109); Sec. 8, 22 (p. 111); Sec. 9, 14, 33, 52 (pp. 114, 117, 122). What is the 'Universe of Discourse'?
XI. Sec. 4, 41 (p. 100); Sec. 5, 71-74 (p. 104); Sec. 6, 31, 32 (p. 107); Sec. 7, 31 (p. 109); Sec. 8, 23 (p. 111); Sec. 9, 15, 34, 53 (pp. 114, 118, 122). What is implied, in a Proposition of Relation, as to the Reality of its Terms?
XII. Sec. 4, 42 (p. 100); Sec. 5, 75-78 (p. 105); Sec. 6, 33 (p. 107); Sec. 7, 32, 33 (pp. 109, 110); Sec. 8, 25 (p. 111); Sec. 9, 16, 35, 54 (pp. 114, 118, 122). Explain the phrase "sitting on the fence".
XIII. Sec. 5, 79-83 (p. 105); Sec. 6, 34, 35 (p. 107); Sec. 7, 34 (p. 110); Sec. 8, 26 (p. 111); Sec. 9, 17, 36, 55 (pp. 114, 118, 122). What are 'Converse' Propositions?
XIV. Sec. 5, 84-88 (p. 105); Sec. 6, 36 (p. 107); Sec. 7, 35, 36 (p. 110); Sec. 8, 27 (p. 111); Sec. 9, 18, 37, 56 (pp. 114, 118, 123). What are 'Concrete' and 'Abstract' Propositions? pg096 XV. Sec. 5, 89-93 (p. 105); Sec. 6, 37, 38 (p. 107); Sec. 7, 37 (p. 110); Sec. 8, 28 (p. 111); Sec. 9, 19, 38, 57 (pp. 115, 118, 123). What is a 'Syllogism'? And what are its 'Premisses' and its 'Conclusion'?
XVI. Sec. 5, 94-97 (p. 106); Sec. 6, 39 (p. 107); Sec. 7, 38, 39 (p. 110); Sec. 8, 29 (p. 111); Sec. 9, 20, 39, 58 (pp. 115, 119, 123). What is a 'Sorites'? And what are its 'Premisses', its 'Partial Conclusions', and its 'Complete Conclusion'?
XVII. Sec. 5, 98-101 (p. 106); Sec. 6, 40 (p. 107); Sec. 7, 40 (p. 110); Sec. 8, 30 (p. 111); Sec. 9, 21, 41, 59, 60 (pp. 115, 119, 124). What are the 'Universe of Discourse', the 'Eliminands', and the 'Retinends', of a Syllogism? And of a Sorites?
pg097
BOOK VIII.
EXAMPLES, ANSWERS, AND SOLUTIONS.
[N.B. Reference tags for Examples, Answers & Solutions will be found in the right margin.]
CHAPTER I.
EXAMPLES.
Sec. 1. EX1
Propositions of Relation, to be reduced to normal form.
1. I have been out for a walk.
2. I am feeling better.
3. No one has read the letter but John.
4. Neither you nor I are old.
5. No fat creatures run well.
6. None but the brave deserve the fair.
7. No one looks poetical unless he is pale.
8. Some judges lose their tempers.
9. I never neglect important business.
10. What is difficult needs attention.
11. What is unwholesome should be avoided.
12. All the laws passed last week relate to excise.
13. Logic puzzles me.
14. There are no Jews in the house.
15. Some dishes are unwholesome if not well-cooked.
16. Unexciting books make one drowsy.
17. When a man knows what he's about, he can detect a sharper.
18. You and I know what we're about.
19. Some bald people wear wigs.
20. Those who are fully occupied never talk about their grievances.
21. No riddles interest me if they can be solved.
pg098 Sec. 2. EX2
Pairs of Abstract Propositions, one in terms of x and m, and the other in terms of y and m, to be represented on the same Triliteral Diagram.
1. No x are m; No m' are y.
2. No x' are m'; All m' are y.
3. Some x' are m; No m are y.
4. All m are x; All m' are y'.
5. All m' are x; All m' are y'.
6. All x' are m'; No y' are m.
7. All x are m; All y' are m'.
8. Some m' are x'; No m are y.
9. All m are x'; No m are y.
10. No m are x'; No y are m'.
11. No x' are m'; No m are y.
12. Some x are m; All y' are m.
13. All x' are m; No m are y.
14. Some x are m'; All m are y.
15. No m' are x'; All y are m.
16. All x are m'; No y are m.
17. Some m' are x; No m' are y'.
18. All x are m'; Some m' are y'.
19. All m are x; Some m are y'.
20. No x' are m; Some y are m.
21. Some x' are m'; All y' are m.
22. No m are x; Some m are y.
23. No m' are x; All y are m'.
24. All m are x; No y' are m'.
25. Some m are x; No y' are m.
26. All m' are x'; Some y are m'.
27. Some m are x'; No y' are m'.
28. No x are m'; All m are y'.
29. No x' are m; No m are y'.
30. No x are m; Some y' are m'.
31. Some m' are x; All y' are m;
32. All x are m'; All y are m.
pg099 Sec. 3. EX3
Marked Triliteral Diagrams, to be interpreted in terms of x and y.
1 . -. (O) . - -. (I) (O) - - - - (O) . - -. (O) . -.
2 . -. (O) . - -. (O) - - - (I) (O) . - -. (O) . -.
3 . -. (O) . - -. (O) (O) - - - - (I) . - -. (O) . -.
4 . -. (O) (O) . - -. (O) - - (I) - (O) . - -. . -.
5 . -. (I) . - -. (O) - - - - (O) . - -. (O) (O) . -.
6 . -. (O) (O) . - -. - - - - (O) . - -. (I) (O) . -.
7 . -. (O) (O) . - -. (I) (O) - - - - (O) . - -. . -.
8 . -. (I) . - -. (O) - - - - (O) (I) . - -. (O) . -.
9 . -. . - -. (O) - - - - (O) (I) . - -. (O) (O) . -.
10 . -. (O) (O) . - -. (O) (I) - - - - (O) . - -. . -.
11 . -. (O) (O) . - -. (I) - - - - (O) . - -. (O) . -.
12 . -. (O) . - -. - (I) - - . - -. (O) (I) . -.
13 . -. (O) (I) . - -. (O) - - - - (O) . - -. (O) . -.
14 . -. (O) (O) . - -. (O) - - - - (O) . - -. (I) . -.
15 . -. . - -. (I) - - - - (O) (O) . - -. . -.
16 . -. (I) . - -. - - - - (O) . - -. (O) (O) . -.
17 . -. (O) . - -. (I) - - - - (O) (O) . - -. (I) (O) . -.
18 . -. (O) (O) . - -. (O) (I) - - - - (O) . - -. (I) . -.
19 . -. (O) (O) . - -. (I) (O) - - - - (O) . - -. (I) . -.
20 . -. (O) . - -. (O) (O) - - - - (I) . - -. (O) . -.
pg100 Sec. 4. EX4
Pairs of Abstract Propositions, proposed as Premisses: Conclusions to be found.
1. No m are x'; All m' are y.
2. No m' are x; Some m' are y'.
3. All m' are x; All m' are y'.
4. No x' are m'; All y' are m.
5. Some m are x'; No y are m.
6. No x' are m; No m are y.
7. No m are x'; Some y' are m.
8. All m' are x'; No m' are y.
9. Some x' are m'; No m are y'.
10. All x are m; All y' are m'.
11. No m are x; All y' are m'.
12. No x are m; All y are m.
13. All m' are x; No y are m.
14. All m are x; All m' are y.
15. No x are m; No m' are y.
16. All x are m'; All y are m.
17. No x are m; All m' are y.
18. No x are m'; No m are y.
19. All m are x; All m are y'.
20. No m are x; All m' are y.
21. All x are m; Some m' are y.
22. Some x are m; All y are m.
23. All m are x; Some y are m.
24. No x are m; All y are m.
25. Some m are x'; All y' are m'.
26. No m are x'; All y are m.
27. All x are m'; All y' are m.
28. All m are x'; Some m are y.
29. No m are x; All y are m'.
30. All x are m'; Some y are m.
31. All x are m; All y are m.
32. No x are m'; All m are y.
33. No m are x; No m are y.
34. No m are x'; Some y are m.
35. No m are x; All y are m.
36. All m are x'; Some y are m.
37. All m are x; No y are m.
38. No m are x; No m' are y.
39. Some m are x'; No m are y.
40. No x' are m; All y' are m.
41. All x are m'; No y are m'.
42. No m' are x; No y are m.
pg101 Sec. 5. EX5
Pairs of Concrete Propositions, proposed as Premisses: Conclusions to be found.
1. I have been out for a walk; I am feeling better.
2. No one has read the letter but John; No one, who has not read it, knows what it is about.
3. Those who are not old like walking; You and I are young.
4. Your course is always honest; Your course is always the best policy.
5. No fat creatures run well; Some greyhounds run well.
6. Some, who deserve the fair, get their deserts; None but the brave deserve the fair.
7. Some Jews are rich; All Esquimaux are Gentiles.
8. Sugar-plums are sweet; Some sweet things are liked by children.
9. John is in the house; Everybody in the house is ill.
10. Umbrellas are useful on a journey; What is useless on a journey should be left behind.
11. Audible music causes vibration in the air; Inaudible music is not worth paying for.
12. Some holidays are rainy; Rainy days are tiresome.
13. No Frenchmen like plumpudding; All Englishmen like plumpudding.
14. No portrait of a lady, that makes her simper or scowl, is satisfactory; No photograph of a lady ever fails to make her simper or scowl.
15. All pale people are phlegmatic; No one looks poetical unless he is pale.
16. No old misers are cheerful; Some old misers are thin.
17. No one, who exercises self-control, fails to keep his temper; Some judges lose their tempers. pg102 18. All pigs are fat; Nothing that is fed on barley-water is fat.
19. All rabbits, that are not greedy, are black; No old rabbits are free from greediness.
20. Some pictures are not first attempts; No first attempts are really good.
21. I never neglect important business; Your business is unimportant.
22. Some lessons are difficult; What is difficult needs attention.
23. All clever people are popular; All obliging people are popular.
24. Thoughtless people do mischief; No thoughtful person forgets a promise.
25. Pigs cannot fly; Pigs are greedy.
26. All soldiers march well; Some babies are not soldiers.
27. No bride-cakes are wholesome; What is unwholesome should be avoided.
28. John is industrious; No industrious people are unhappy.
29. No philosophers are conceited; Some conceited persons are not gamblers.
30. Some excise laws are unjust; All the laws passed last week relate to excise.
31. No military men write poetry; None of my lodgers are civilians.
32. No medicine is nice; Senna is a medicine.
33. Some circulars are not read with pleasure; No begging-letters are read with pleasure.
34. All Britons are brave; No sailors are cowards.
35. Nothing intelligible ever puzzles me; Logic puzzles me.
36. Some pigs are wild; All pigs are fat. pg103 37. All wasps are unfriendly; All unfriendly creatures are unwelcome.
38. No old rabbits are greedy; All black rabbits are greedy.
39. Some eggs are hard-boiled; No eggs are uncrackable.
40. No antelope is ungraceful; Graceful creatures delight the eye.
41. All well-fed canaries sing loud; No canary is melancholy if it sings loud.
42. Some poetry is original; No original work is producible at will.
43. No country, that has been explored, is infested by dragons; Unexplored countries are fascinating.
44. No coals are white; No niggers are white.
45. No bridges are made of sugar; Some bridges are picturesque.
46. No children are patient; No impatient person can sit still.
47. No quadrupeds can whistle; Some cats are quadrupeds.
48. Bores are terrible; You are a bore.
49. Some oysters are silent; No silent creatures are amusing.
50. There are no Jews in the house; No Gentiles have beards a yard long.
51. Canaries, that do not sing loud, are unhappy; No well-fed canaries fail to sing loud.
52. All my sisters have colds; No one can sing who has a cold.
53. All that is made of gold is precious; Some caskets are precious.
54. Some buns are rich; All buns are nice.
55. All my cousins are unjust; All judges are just. pg104 56. Pain is wearisome; No pain is eagerly wished for.
57. All medicine is nasty; Senna is a medicine.
58. Some unkind remarks are annoying; No critical remarks are kind.
59. No tall men have woolly hair; Niggers have woolly hair.
60. All philosophers are logical; An illogical man is always obstinate.
61. John is industrious; All industrious people are happy.
62. These dishes are all well-cooked; Some dishes are unwholesome if not well-cooked.
63. No exciting books suit feverish patients; Unexciting books make one drowsy.
64. No pigs can fly; All pigs are greedy.
65. When a man knows what he's about, he can detect a sharper; You and I know what we're about.
66. Some dreams are terrible; No lambs are terrible.
67. No bald creature needs a hairbrush; No lizards have hair.
68. All battles are noisy; What makes no noise may escape notice.
69. All my cousins are unjust; No judges are unjust.
70. All eggs can be cracked; Some eggs are hard-boiled.
71. Prejudiced persons are untrustworthy; Some unprejudiced persons are disliked.
72. No dictatorial person is popular; She is dictatorial.
73. Some bald people wear wigs; All your children have hair.
74. No lobsters are unreasonable; No reasonable creatures expect impossibilities. pg105 75. No nightmare is pleasant; Unpleasant experiences are not eagerly desired.
76. No plumcakes are wholesome; Some wholesome things are nice.
77. Nothing that is nice need be shunned; Some kinds of jam are nice.
78. All ducks waddle; Nothing that waddles is graceful.
79. Sandwiches are satisfying; Nothing in this dish is unsatisfying.
80. No rich man begs in the street; Those who are not rich should keep accounts.
81. Spiders spin webs; Some creatures, that do not spin webs, are savage.
82. Some of these shops are not crowded; No crowded shops are comfortable.
83. Prudent travelers carry plenty of small change; Imprudent travelers lose their luggage.
84. Some geraniums are red; All these flowers are red.
85. None of my cousins are just; All judges are just.
86. No Jews are mad; All my lodgers are Jews.
87. Busy folk are not always talking about their grievances; Discontented folk are always talking about their grievances.
88. None of my cousins are just; No judges are unjust.
89. All teetotalers like sugar; No nightingale drinks wine.
90. No riddles interest me if they can be solved; All these riddles are insoluble.
91. All clear explanations are satisfactory; Some excuses are unsatisfactory.
92. All elderly ladies are talkative; All good-tempered ladies are talkative.
93. No kind deed is unlawful; What is lawful may be done without scruple. pg106 94. No babies are studious; No babies are good violinists.
95. All shillings are round; All these coins are round.
96. No honest men cheat; No dishonest men are trustworthy.
97. None of my boys are clever; None of my girls are greedy.
98. All jokes are meant to amuse; No Act of Parliament is a joke.
99. No eventful tour is ever forgotten; Uneventful tours are not worth writing a book about.
100. All my boys are disobedient; All my girls are discontented.
101. No unexpected pleasure annoys me; Your visit is an unexpected pleasure.
Sec. 6. EX6
Trios of Abstract Propositions, proposed as Syllogisms: to be examined.
1. Some x are m; No m are y'. Some x are y.
2. All x are m; No y are m'. No y are x'.
3. Some x are m'; All y' are m. Some x are y.
4. All x are m; No y are m. All x are y'.
5. Some m' are x'; No m' are y. Some x' are y'.
6. No x' are m; All y are m'. All y are x'.
7. Some m' are x'; All y' are m'. Some x' are y'.
8. No m' are x'; All y' are m'. All y' are x.
9. Some m are x'; No m are y. Some x' are y'.
10. All m' are x'; All m' are y. Some y are x'.
11. All x are m'; Some y are m. Some y are x'.
12. No x are m; No m' are y'. No x are y'.
13. No x are m; All y' are m. All y' are x'.
14. All m' are x'; All m' are y. Some y are x'.
15. Some m are x'; All y are m'. Some x' are y'.
16. No x' are m; All y' are m'. Some y' are x.
17. No m' are x; All m' are y'. Some x' are y'. pg107 18. No x' are m; Some m are y. Some x are y.
19. Some m are x; All m are y. Some y are x'.
20. No x' are m'; Some m' are y'. Some x are y'.
21. No m are x; All m are y'. Some x' are y'.
22. All x' are m; Some y are m'. All x' are y'.
23. All m are x; No m' are y'. No x' are y'.
24. All x are m'; All m' are y. All x are y.
25. No x are m'; All m are y. No x are y'.
26. All m are x'; All y are m. All y are x'.
27. All x are m; No m are y'. All x are y.
28. All x are m; No y' are m'. All x are y.
29. No x' are m; No m' are y'. No x' are y'.
30. All x are m; All m are y'. All x are y'.
31. All x' are m'; No y' are m'. All x' are y.
32. No x are m; No y' are m'. No x are y'.
33. All m are x'; All y' are m. All y' are x'.
34. All x are m'; Some y are m'. Some y are x.
35. Some x are m; All m are y. Some x are y.
36. All m are x'; All y are m. All y are x'.
37. No m are x'; All m are y'. Some x are y'.
38. No x are m; No m are y'. No x are y'.
39. No m are x; Some m are y'. Some x' are y'.
40. No m are x'; Some y are m. Some x are y.
Sec. 7. EX7
Trios of Concrete Propositions, proposed as Syllogisms: to be examined.
1. No doctors are enthusiastic; You are enthusiastic. You are not a doctor.
2. Dictionaries are useful; Useful books are valuable. Dictionaries are valuable.
3. No misers are unselfish; None but misers save egg-shells. No unselfish people save egg-shells.
4. Some epicures are ungenerous; All my uncles are generous. My uncles are not epicures. pg108 5. Gold is heavy; Nothing but gold will silence him. Nothing light will silence him.
6. Some healthy people are fat; No unhealthy people are strong. Some fat people are not strong.
7. "I saw it in a newspaper." "All newspapers tell lies." It was a lie.
8. Some cravats are not artistic; I admire anything artistic. There are some cravats that I do not admire.
9. His songs never last an hour; A song, that lasts an hour, is tedious. His songs are never tedious.
10. Some candles give very little light; Candles are meant to give light. Some things, that are meant to give light, give very little.
11. All, who are anxious to learn, work hard; Some of these boys work hard. Some of these boys are anxious to learn.
12. All lions are fierce; Some lions do not drink coffee. Some creatures that drink coffee are not fierce.
13. No misers are generous; Some old men are ungenerous. Some old men are misers.
14. No fossil can be crossed in love; An oyster may be crossed in love. Oysters are not fossils.
15. All uneducated people are shallow; Students are all educated. No students are shallow.
16. All young lambs jump; No young animals are healthy, unless they jump. All young lambs are healthy.
17. Ill-managed business is unprofitable; Railways are never ill-managed. All railways are profitable.
18. No Professors are ignorant; All ignorant people are vain. No professors are vain. pg109 19. A prudent man shuns hyaenas; No banker is imprudent. No banker fails to shun hyaenas.
20. All wasps are unfriendly; No puppies are unfriendly. Puppies are not wasps.
21. No Jews are honest; Some Gentiles are rich. Some rich people are dishonest.
22. No idlers win fame; Some painters are not idle. Some painters win fame.
23. No monkeys are soldiers; All monkeys are mischievous. Some mischievous creatures are not soldiers.
24. All these bonbons are chocolate-creams; All these bonbons are delicious. Chocolate-creams are delicious.
25. No muffins are wholesome; All buns are unwholesome. Buns are not muffins.
26. Some unauthorised reports are false; All authorised reports are trustworthy. Some false reports are not trustworthy.
27. Some pillows are soft; No pokers are soft. Some pokers are not pillows.
28. Improbable stories are not easily believed; None of his stories are probable. None of his stories are easily believed.
29. No thieves are honest; Some dishonest people are found out. Some thieves are found out.
30. No muffins are wholesome; All puffy food is unwholesome. All muffins are puffy.
31. No birds, except peacocks, are proud of their tails; Some birds, that are proud of their tails, cannot sing. Some peacocks cannot sing.
32. Warmth relieves pain; Nothing, that does not relieve pain, is useful in toothache. Warmth is useful in toothache. pg110 33. No bankrupts are rich; Some merchants are not bankrupts. Some merchants are rich.
34. Bores are dreaded; No bore is ever begged to prolong his visit. No one, who is dreaded, is ever begged to prolong his visit. |
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